A Virtual Element Method for the Transmission Eigenvalue Problem
It provides a novel numerical method for solving transmission eigenvalue problems, which are important in inverse scattering, but the contribution is incremental as it extends existing VEM techniques to a specific problem class.
The paper develops a virtual element method for a non-selfadjoint fourth-order eigenvalue problem from transmission eigenvalue problems, achieving optimal error estimates for eigenfunctions and double order for eigenvalues, validated by numerical experiments.
In this paper, we analyze a virtual element method (VEM) for solving a non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a $C^1$-conforming discretization by means of the VEM. We use the classical approximation theory for compact non-selfadjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.